Anomalous Josephson effect in d -wave superconductor junctions on a topological insulator surface

We study the Josephson effect of d-wave superconductor (DS)/ferromagnet insulator (FI)/DS junctions on a surface of topological insulator (TI). We calculate Josephson current Iφ for various orientations of the junctions where φ is the macroscopic phase difference between two DSs. In certain configurations, we find anomalous current-phase relation I(φ)=-I-φ+π with 2π periodicity. In the case where the first-order Josephson coupling is absent without magnetization in FI, I(φ) can be proportional to cosφ. In symmetric junctions, the magnitude of the obtained Josephson current is enhanced due to the zero-energy states on the edge of DS on TI. Even if we introduce an s-wave component of pair potential in DS, we can still expect the anomalous current-phase relation in asymmetric DS junctions with Iφ=0≠0. This can be used to probe the induced d-wave component of pair potential on a TI surface in high-Tc cuprate/TI hybrid structures.